Let \(p(x,y)\), \(q(x,y)\) be polynomials with real or complex coefficients and let \(F=(p,q)\) be a corresponding polynomial mapping from \(\mathbb{R}^2(\mathbb{C}^2)\) to itself. We will denote by \(J(p,q)\) the Jacobian of \(F\), i.e., \[ J(p,q)= {\partial p\over\partial x} {\partial q\over\partial y}-{\partial p\over\partial y} {\partial q\over\partial x}. \] The classical Jacobian problem (conjecture) is to show that if \(J(p,q)=1\), then \(F\) is invertible. This conjecture was first posed by O. H. Keller [Monatsh. Math. Phys. 47, 299-306 (1939; Zbl 0021.15303)] and after more than 50 years it still remains an open problem. A survey of a number of the partial results and a historical account one can find in the paper of H. Bass, E. H. Connell and D. Wright [Bull. Am. Math. Soc., New Ser. 7, 287-330 (1982; Zbl 0539.13012)]. It is worth to notice that in most of the papers on the Jacobian conjecture, the authors tried to prove it and only few of them discussed the possibility of a counterexample. We want to mention the paper of A. G. Vitushkin [Proc. int. Conf. Manifold, rel. Top. Topol., Tokyo 1973, 415-417 (1975; Zbl 0309.14010)], who presented some topological arguments in favor of a negative solution of the Jacobian conjecture.
In this paper, we construct a counterexample to the so-called real Jacobian conjecture (see, for example, J. D. Randall [Proc. Sympos. Pure Math. 40, Part 2, 411-414 (1983; Zbl 0524.26009)]), which is stronger than the classical one and asks whether a polynomial mapping \(F:\mathbb{R}^2\to\mathbb{R}^2\) with a nonvanishing Jacobian \(J(F)\) is a global diffeomorphism from \(\mathbb{R}^2\) onto \(\mathbb{R}^2\).